Seminars Sorted by Series

Given a connected reductive group G and a Levi subgroup M, Braverman-Gaistgory and Laumon constructed geometric Eisenstein functors which take Hecke eigensheaves on the moduli stack $Bun_{M}$ of M-bundles on a curve to eigensheaves on the moduli...

We study spectral invariants of torsion type on Calabi-Yau manifolds of dimension three. We define a new holomorphic invariant modifying the so-called BCOV torsion and explore its applications in the study of moduli space of Calabi-Yau manifolds...

If we deform a submanifold along its mean curvature vector, which is called mean curvature flow (MCF), the area will decrease most rapidly. Being Lagrangian is preserved along MCF in Kahler-Einstein manifolds. We can thus try to construct special...

Let $\Omega$ and $\tilde{\Omega}$ be domains in the hyperbolic plane with smooth boundary. Assume that both domains are uniformly convex, and have the same area. We show that there exists an area-preserving, orientation-preserving diffeomorphism $f...

This is joint work with Y. Canzani, B. Clarke, N. Kamran, L. Silberman and J. Taylor. We construct Gaussian measure on the manifold of Riemannian metrics with the fixed volume form. We show that diameter and Laplace eigenvalue and volume entropy...

We introduce the term `superconcentration' to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens...

We discuss recently established criteria for the formation of extended states on tree graphs in the presence of disorder. These criteria have the surprising implication that for bounded random potentials, as in the Anderson model, in the weak...

It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent \chi and the wandering exponent \xi are related through the universal relation \chi=2\xi -1, irrespective of...

We consider a passive scalar field under the action of pumping, diffusion and advection by a smooth flow with a Lagrangian chaos. We present theoretical arguments showing that scalar statistics is not conformal invariant and formulate new effective...

We consider asymptotics of the correlation functions of characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$ and the hermitian sample covariance matrices $X_n=n^{-1}A_{m,n}^*A_{m,n}$. We use the integration over the...

In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of...

A proposal for QED in half space is made. Starting from the well known principle of mirror charges in electrostatics, we formulate boundary conditions for electromagnetic fields and charge carrying currents both in the classical and the quantum...

The Kuramoto synchronization model is the reference model for synchronization phenomena in biology (and, to a certain extent, also in other fields). The model is formulated as a dynamical system of interacting plane rotators. Variations of it...

Macdonald processes are probability measures on sequences of partitions de?ned in terms of nonnegative specializations of the Macdonald symmetric functions and two parameters q, t ? [0,1). Utilizing the Macdonald difference operators we prove...

In 2002, J.B. Conrey and K. Soundararajan showed that there are infinitely many Dirichlet $L$-functions which do not vanish on the critical segment. Let $\mathcal{G}$ be the family of Dirichlet $L$-functions they considered. Following their work, a...

Suppose E is an elliptic curve defined over a number field K, and p is a prime where E has good ordinary reduction. The usual methods of Iwasawa theory give a single Iwasawa module from which one can recover the Selmer groups of E over all finite...

In this talk, I will give applications of Kedlaya's recent results on p-adic differential equations. In particular, I will give a new proof of Colmez-Fontaine's theorem which describes semistable p-adic representations.

For a number field K and a (compact) p-adic Lie groups G, the inverse Galois problem asks whether G can be realized as the Galois group of an extension of K. Already in the case that G is zero-dimensional, this is too difficult. So I propose to...

In this talk, I'd like to describe my recent work with P.Sarnak concerning the equidistribution properties for Hecke igenforms on the modular surface. We evaluate asymptotically the variance for the equidistribution by means of the trace formula and...

Serre conjectured that all continuous, irreducible, odd $\rho:G_{\mathbf{Q}} \to \mathrm{GL}_2(\overline{\mathbf{F}}_p)$ arise from modular forms. If $\rho$ is modular, then proven refinements provide recipes for the possible weights and levels of...

Let $f$ be a primitive cusp form of weight at least 2, and let $\rho_f$ be the $p$-adic Galois representation attached to $f$. If $f$ is $p$-ordinary, then it is known that the restriction of $\rho_f$ to a decomposition group at $p$ is `upper...

It is well-known that to study the space of modular forms of a fixed weight and level over $Q_P$, it is useful to embed this finite-dimensional space in a $p$-adic Banach space of overconvergent modular forms, and use methods of $p$-adic analysis...

After a brief review of the theory of multiple zeta values and zeta functions, we will discuss the multiple Hurwitz zeta function given by $$\zeta(s_1, s_2,..., s_r; x_1, x_2,..., x_r) = \sum_{n_1>n_2>\cdots >n_r\geq 1} {1 \over (n_1+x_1)^{s_1} (n_2...

In this talk, I will explain a new result on the distribution of rational points of bounded height on "wonderful" compactifications of semi-simple groups of adjoint type. A special case of the result is the conjecture of Manin on the distribution of...

Let K be an imaginary quadratic field. Modular forms for K are related to the cohomology of arithmetic 3-manifolds. By using the Galois representations associated to such forms we produce an explicit tower of rational homology three spheres with...

I will discuss relatively new kinds of zeta and L-functions - the Ihara-Selberg-Artin L-functions attached to finite graph coverings. In joint work with Harold Stark, we have found that many of the methods used by number theorists in investigations...

Let E be an elliptic curve over the rationals and p a fixed prime. A (very hard) question of Kolyvagin asks if there exists a quadratic discriminant d such that the Tate-Shafarevich group of the quadratic twist E_d has order prime to p. I will...

Shimura Curves over Q parameterize abelian surfaces with quaternionic multiplication. However the universal families exist only when considerable level structure is added. This makes it difficult to write eqautions for these families, or...

The title refers to a conjecture that Serre made in the early 1970's, that has proved to be very influential. I will sketch the main ideas of the proof of the level 1 case of Serre's conjecture. If time permits, I will indicate how these ideas can...

We will talk about a proof of Manin's conjecture on the asymptotic density of rational points of bounded height for the special case of a compactification of a semisimple algebraic group defined over a number field. The main tool is the mixing...

In his last letter to Hardy, Ramanujan defined 17 peculiar functions which are now referred to as his mock theta functions. Although these mysterious functions have been investigated by many mathematicians over the years, many of their most basic...

In this lecture, I will describe some results from the thesis of Jarad Schofer, (Maryland, 2005), which provide a generalization of the Gross-Zagier factorization of singular moduli or arbitrary Borcherds forms. After a review of the construction of...

I will describe some identities of linear forms and distributions attached to representations in theta correspondence. These identities can be used to prove formula of special values of L-functions.

In this talk I will discuss a particular family of modular units constructed using functional solutions to q-difference equations found in the work of Selberg. Arising in this way, these objects are of interest for various analytic properties and...